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On a Theorem of Heilbronn Concerning the Fractional Part of θn2

Published online by Cambridge University Press:  20 November 2018

Ming-Chit Liu
Ming-Chit Liu*
Affiliation:
University of Hong Kong, Pokfulum Road, Hong Kong
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1. In 1948 Heilbronn [4] proved the following theorem.

THEOREM H. For every real θ and every positive integer N, there is an integer n satisfying

(1.1)

whereis an arbitrarily small number,depends only on, and ‖t‖ means the distance from t to the nearest integer.

The interest of the result (1.1) is that the inequality is uniform in θ, and is therefore analogous to the classical inequality of Dirichlet for the fractional part of θn. In this paper we shall prove the following theorem.

THEOREM. For every real θ and every positive integer N, there is an integer n satisfying

(1.2)

where A is an absolute constant and.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 784 - 788
Copyright
Copyright © Canadian Mathematical Society 1970

References

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